Precalculus Examples

$2 \log_{3} (x) = 3 \log_{3} (5)$

Step 1

Simplify.

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Step 1.1

Simplify $2 \log_{3} (x)$ by moving $2$ inside the logarithm.

$\log_{3} (x^{2}) = 3 \log_{3} (5)$

Step 1.2

Simplify $3 \log_{3} (5)$ by moving $3$ inside the logarithm.

$\log_{3} (x^{2}) = \log_{3} (5^{3})$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$x^{2} = 5^{3}$

Step 3

Solve for $x$ .

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Step 3.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{5^{3}}$

Step 3.2

Simplify $\pm \sqrt{5^{3}}$ .

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Step 3.2.1

Raise $5$ to the power of $3$ .

$x = \pm \sqrt{125}$

Step 3.2.2

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 3.2.2.1

Factor $25$ out of $125$ .

$x = \pm \sqrt{25 (5)}$

Step 3.2.2.2

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2} \cdot 5}$

Step 3.2.3

Pull terms out from under the radical.

$x = \pm 5 \sqrt{5}$

Step 3.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 5 \sqrt{5}$

Step 3.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 5 \sqrt{5}$

Step 3.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 5 \sqrt{5}, - 5 \sqrt{5}$

Step 4

Exclude the solutions that do not make $2 \log_{3} (x) = 3 \log_{3} (5)$ true.

$x = 5 \sqrt{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 5 \sqrt{5}$

Decimal Form:

$x = 11.18033988 \dots$